Linear series with cusps and $n$-fold points

Abstract:

A linear series $(V, \mathcal {L})$ on a curve $X$ has an $n$-fold point along a divisor $D$ of degree $n$ if $\dim (V \cap {H^0}(X, \mathcal {L}( - D))) \geqslant \dim (V) - 1$. The linear series has a cusp of order $e$ at a point $P$ if $\dim (V \cap {H^0}(X, \mathcal {L}( - (e + 1)P))) \geqslant \dim (V) - 1$. Linear series with cusps and $n$-fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.